

A129533


Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= n.


6



0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 6, 12, 6, 0, 0, 10, 30, 30, 10, 0, 0, 15, 60, 90, 60, 15, 0, 0, 21, 105, 210, 210, 105, 21, 0, 0, 28, 168, 420, 560, 420, 168, 28, 0, 0, 36, 252, 756, 1260, 1260, 756, 252, 36, 0, 0, 45, 360, 1260, 2520, 3150, 2520, 1260, 360, 45, 0, 0, 55, 495
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OFFSET

0,8


COMMENTS

Previous name was: Triangle read by rows: T(n,k)=derivative of the qbinomial coefficient [n,k] evaluated at q=1 (0<=k<=n).  N. J. A. Sloane, Jan 06 2016
For example, T(5,2)=30 because [5,2] = q^6 + q^5 + 2*q^4 + 2*q^3 + 2*q^2 + q + 1 with derivative 6q^5 + 5q^4 + 8q^3 + 6q^2 + 4q + 1, having value 30 at q=1.  Emeric Deutsch, Apr 22 2007
Sum of entries in nth antidiagonal = n(n1)2^(n3) = A001788(n1).
T(n,k) = A094305(n2, k1) for n >= 2, k >= 1.
T(n,k) is total number of pips on a set of generalized linear dominoes with n cells (rather than two) and with the number of pips in each cell running from 0 to k (rather than 6). T(2,6) = 168 gives the total number of pips on a standard set of dominoes. We regard a generalized linear domino with n cells and up to k pips per cell as an ordered ntuple [i_1, i_2, ..., i_n] with 0 <= i_1 <= i_2 <= ... <= i_n <= k.  Alan Shore and N. J. A. Sloane, Jan 06 2016
T(n,k) can also be written more symmetrically as the trinomial coefficient (n+k; n1, k1, 2).  N. J. A. Sloane, Jan 06 2016
As a triangle read by rows, T(n,k) is the total number of inversions over all length n binary words having exactly k 1's. T(n,k) is also the total area above all North East lattice paths from the origin to the point (k,nk).  Geoffrey Critzer, Mar 22 2018


REFERENCES

G. E. Andrews, The Theory of Partitions, AddisonWesley, 1976.


LINKS



FORMULA

T(n,k) = (1/2)*k*(k+1)*binomial(n,k+1).
G.f.: G(q,z) = qz^2/(1zqz)^3.


EXAMPLE

Array begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... (A000004)
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, ... (A000217)
0, 3, 12, 30, 60, 105, 168, 252, 360, 495, 660, 858, ... (A027480)
0, 6, 30, 90, 210, 420, 756, 1260, 1980, 2970, 4290, ... (A033487)
0, 10, 60, 210, 560, 1260, 2520, 4620, 7920, 12870, ... (A266732)
0, 15, 105, 420, 1260, 3150, 6930, 13860, 25740, ... (A240440)
0, 21, 168, 756, 2520, 6930, 16632, 36036, ... (A266733)
...
If regarded as a triangle, this begins:
0;
0, 0;
0, 1, 0;
0, 3, 3, 0;
0, 6, 12, 6, 0;
0, 10, 30, 30, 10, 0;
0, 15, 60, 90, 60, 15, 0;
...


MAPLE

dd:=proc(n, m) if m=0 or n=0 then 0 else (m+n)!/(2*(m1)!*(n1)!); fi; end;
f:=n>[seq(dd(n, m), m=0..30)];
for n from 0 to 10 do lprint(f(n)); od: # produces sequence as square array
T:=(n, k)>k*(k+1)*binomial(n, k+1)/2: for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form


MATHEMATICA

Table[Table[D[Expand[FunctionExpand[QBinomial[n, k, q]]], q] /. q > 1, {k, 0, n}], {n, 0, 15}] // Grid (* Geoffrey Critzer, Mar 22 2018 *)


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STATUS

approved



